222(g) unlawful presence doesn't apply to her as a minor.
If the child was born out of wedlock, there has to be a bonafide parent-child relationship (providing support or living together.)
As the minor child of a USC, there's absolutely no legal reason she has to depart the US - the OP should get another opinion from an AILA lawyer.
Children
General
The meaning of "child" is not as simple as it first appears. For the purposes of family-based immigration, a "child" is defined in INA §101(b)(1) as follows:
a child born in wedlock;
a stepchild whether or not born out of wedlock, provided the child had not reached the age of eighteen years at the time the marriage creating the status of stepchild occurred;
a child legitimated under the law of the child's residence or domicile, or under the law of father's residence or domicile, whether in or outside the United States, if such legitimation takes place before the child reaches the age of eighteen years and the child is in the legal custody of the legitimating parent or parents at the time of such legitimation;
a child born out of wedlock, by, through whom, or on whose behalf a status, privilege, or benefit is sought by virtue of the relationship of the child to its natural mother or to its natural father if the father has or had a bona fide parent-child relationship with the person;
a child adopted while under the age of sixteen years if the child has been in the legal custody of, and has resided with, the adopting parent or parents for at least two years: provided that no natural parent of any such adopted child shall thereafter, by virtue of such parentage, be accorded any right, privilege, or status under this Act; or
a child, under the age of sixteen at the time an immediate relative petition is filed on his or her behalf, who is an orphan because of the death or disappearance of, abandonment or desertion by, or separation or loss from, both parents, or for whom the sole or surviving parent is incapable of providing the proper care and has in writing irrevocably released the child for emigration and adoption; who has been adopted abroad by a United States citizen and spouse jointly, or by an unmarried United States citizen at least twenty-five years of age, who personally saw and observed the child prior to or during the adoption proceedings; or who is coming to the United States for adoption by a United States citizen and spouse jointly, or by an unmarried United States citizen at least twenty-five years of age, who have or has complied with the proadoption requirements, if any, of the child's proposed residence: provided that the Attorney General is satisfied that proper care will be furnished the child if admitted to the United States: Provided further, that no natural parent or prior adoptive parent of any such child shall thereafter, by virtue of such parentage, be accorded any right, privilege, or status under this Act.
The parent-child relationship must continue to exist at the time that immigration benefits are sought. A child includes only an unmarried person under the age of 21. Accordingly, the child must be both unmarried and under 21 at the time the visa is issued or Adjustment of status and at the time that he or she applies for entry to the United States. If the child marries or becomes 21 after the visa is issued and before he or she applies for entry, he or she becomes disqualified for immediate relative status. However, certain exceptions to this "age-out" problem exist as a result of the Child Status Protection Act, which is discussed elsewhere in this web site.